Simplify the following expression and state the condition under which the simplification is valid. You can assume that $x \neq 0$. $y = \dfrac{2}{x(4x - 9)} \div \dfrac{-10}{24x^2 - 54x} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $y = \dfrac{2}{x(4x - 9)} \times \dfrac{24x^2 - 54x}{-10} $ When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ 2 \times (24x^2 - 54x) } { x(4x - 9) \times -10 } $ $ y = \dfrac {2 \times 6x(4x - 9)} {-10 \times x(4x - 9)} $ $ y = \dfrac{12x(4x - 9)}{-10x(4x - 9)} $ We can cancel the $4x - 9$ so long as $4x - 9 \neq 0$ Therefore $x \neq \dfrac{9}{4}$ $y = \dfrac{12x \cancel{(4x - 9})}{-10x \cancel{(4x - 9)}} = -\dfrac{12x}{10x} = -\dfrac{6}{5} $